Math 2002
Section 1.02: iterated integrals
R. Milson
Winter 2023
Authored by R. Milson (robert.milson@dal.ca) Do not use or distirbute without the author’s permission. 1
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Iterated Integrals and Cartesian Products, Lecture notes of Mathematics
An introduction to the concept of iterated integrals, which are used to evaluate double integrals over rectangular domains. It covers the theorem on iterated integrals, several examples demonstrating the evaluation of iterated integrals, the concept of cartesian products, and the property of product integrals. Aimed at students studying multivariable calculus and is likely to be useful as study notes, lecture notes, or for exam preparation. The content covers fundamental topics in advanced calculus and could be relevant to university-level courses in mathematics, physics, or engineering.
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2022/2023
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Math 2002
Section 1.02: iterated integrals
R. Milson Winter 2023
Outline
▶ (^) Text: section 5. ▶ (^) Supplemental videos (mathispower4u.com): Integrating Functions of Two Variables ▶ (^) Iterated integrals ▶ (^) Cartesian product ▶ (^) Product integrals
Iterated integrals
▶ (^) Theorem. Let R be the rectangle a ≤ x ≤ b, c ≤ y ≤ d and f (x, y ) a continuous function defined on R. Then, Z Z
R
f (x, y )dA =
Z (^) b
a
Z (^) d
c
f (x, y )dy
dx =
Z (^) d
c
Z (^) b
a
f (x, y )dx
dy.
▶ (^) The inner integrals above are examples of ”partial integration”. Just like partial derivatives, the integration is performed with respect to one variable, and all other variables are treated like constants.
Example 1
▶ (^) Evaluate
Z Z
R
(x + y )dA over the rectangle 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1.
▶ (^) Solution. We evaluate the double integral by re-expressing it as an interated integral: Z Z
R
(x + y )dA =
Z 1
0
Z 1
0
(x + y )dx
dy
Z 1
0
x^2 2
- xy
x=
x=
dy
Z 1
0
- y
dy
y 2
y 2 2
y = y =
▶ (^) Note: integrating w.r.t. y first, and x second gives the same answer. ▶ (^) Note: the above integral represents the volume of a 1 × 1 × 2 box sliced in half along the diagonal, and so, as expected has a volume value of 2 ÷ 1 = 1.
Cartesian products
▶ (^) Let A, B be sets of numbers. In set theory the cartesian product A × B is defined to be the set of pairs
A × B = {(a, b) : a ∈ A, b ∈ B}
▶ (^) For a < b, the notation [a, b] refers to the closed interval
[a, b] = {x : a ≤ x ≤ b}.
▶ (^) Cartesian products of intervals provide an alternative, compact notation for rectangles. The symbol [a, b] × [c, d] refers to the rectangle corresponding to a ≤ x ≤ b, c ≤ y ≤ d.
Example 3
▶ (^) Calculate I =
RR
R y^ sin(xy^ )dA^ where^ R^ = [1,^ 2]^ ×^ [0, π]. ▶ (^) Solution. Integrating with respect to x first gives
I =
Z (^) y =π
y =
Z (^) x=
x=
y sin(xy )dx
dy =
Z (^) π
0
[− cos(xy )]x x=2=
Z (^) π
0
(− cos(2y ) + cos(y ))dy = −
sin 2y + sin y
y =π y =
▶ (^) Note: integrating w.r.t. y first is problematic, because it results in an complicated anti-derivative.
Example 4
▶ (^) Evaluate I =
RR
R sin^ x^ cos^ ydA^ with^ R^ = [0, π/2]^ ×^ [0, π/2]. ▶ (^) Applying the product formula:
I =
Z (^) π/ 2
0
sin xdx
Z (^) π/ 2
0
cos ydy
h − cos x
ix=π/ 2 x=
h sin y
iy =π/ 2 y = = (−0 + 1) × (1 − 0) = 1.